# find a, b and c in the following sine function: y=a sin(bx + c)so that: * the maximum value of y is 6 when x = 0, y = 6 * the period of the graph is equal to pi * a, b and c are positive and c is...

find a, b and c in the following sine function:

y=a sin(bx + c)

so that:

* the maximum value of y is 6 when x = 0, y = 6

* the period of the graph is equal to pi

* a, b and c are positive and c is less than 2pi

I'd really appreciate any help with this question.

Thanks

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

We want a sinusoid in the form `y=asin(bx+c)` such that the maximum occurs at (0,6), the period is `pi` , and the constants `a,b,c>0;c<2pi` :

Examine `Asin(B(x-h))+k` :

A affects the amplitude (max and min) (if A<0 it reflects the base graph across the x-axis -- not needed here since we want A>0)

B affects the period -- the period is found by `p=(2pi)/B`

h,k are horizontal and vertical translations respectively. For this problem there is no vertical translation, so we are interested in h.

(1) The maximum is 6; the maximum for `sinx` is 1, so A=6

(2) The period `p=pi` , but `p=pi=(2pi)/b==>B=2`

(3) Since the maximum occurs at x=0 we compare to the base function `sinx` where the maximum occurs at `pi/2` . We need a horizontal shift left `pi/2` units.

Putting it all together we get `y=6sin2(x+pi/4)` (Note that since we want a horizontal shift of `pi/2` units, and b=2 we have `h=-pi/4,x-h=x+pi/4` )

Rewriting in the form `y=asin(bx+c)` we get `y=6sin(2x+pi/2)`

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The equation we seek is `y=6sin(2x+pi/2)` so `a=6,b=2,c=pi/2`

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The graph of `y=6sin(2x+pi/2)` :

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